Chapter $7$ Assignment: Time Value of Money $($TVM$)$Instructions:For each Time Value of Money $($TVM$)$ problem, follow these steps:$1.$ Clearly write down each variable name, including its unit of measurement $($e$.$g$.,$ dollars, percent, years$)\mathrm{.}2.$ Perform the necessary calculations$\mathrm{.}3.$ Ensure that your answer includes the correct units$\mathrm{.}1.(5*2=10$ points$)$ Jan is considering purchasing a new car for $$40,000.$ The dealer is offering two options on the purchase: Option $1$: Receive a $$5,000$ rebate on the price of the car and finance the balance over $5$ years at $4\%$ interest, Option $2$: Finance the vehicle for $6$ years at $0\%$ interest, but no rebate.a$)$ What would Jans monthly payments be if she elects option #$1?$ What Is the Total Cost of the loan?b$)$ What would Jans monthly payments be if she elects option #$2?$ What Is the Total Cost of the loan? Which option she should go for?$($Hint: Calculate the total cost by multiplying the monthly payment by the loan term. Choose the option with the lower total cost.$)2.(5$ points$)$ Big Money Bob won $$50$ million in the New York lottery. He can elect to receive a single lump$-$sum payout of $$22$ million after taxes or receive an annuity of $$1,500,000$ after tax, at the end of each year for the next $20$ years. What rate of return would he need to earn to make the lump$-$sum payout, equivalent to the annuity payment? Should Bob take the lump$-$sum payment or the annuity?$($Hint: To find out what interest rate Big Money Bob needs to earn for his lump$-$sum payment to be as valuable as the annual annuity payments, focus on determining the rate that makes the future cash flows from the annuity equal to the current cash amount of the lump sum.$)3.(5$ points$)$ Sarah is considering investing in a new project for her software development company. The initial cost of the software they are developing is $$8,000.$ She expects the following cash flows for the next three years at the end of each year:Year $1$: $$3,000$Year $2$: $$2,500$Year $3$: $$3,200$Assume the software can be sold for $$1,000$ at the end of year $3$ and Davids required rate of return is $8\%.$Calculate the Net Present Value $($NPV$)$ of the project and advise her on whether she should proceed with the investment$4.(5*2=10$ points$)$ You bought a house for $$425,000.$ You made a down payment of $20\%$and financed the balance over $30$ years at $3\mathrm{.}5\%$ annual interest. a$)$ How much will your monthly payments be$?$ b$)$ How much principal and interest will you pay on the first year assuming your first payment is due on January $1?$c$)$ How much principal and interest will you pay on the $24$th month?$5.(5$ points$)$ James deposits $$20,000$ in a savings account that earns $5\%$ interest, compounded quarterly. He plans to leave the money for $5$ years. How much will James have in his account after $5$ years?$6.(5$ points$)$ Lucy deposits $$200$ into a savings account for five years at the end of each year, earning $6\%$ interest annually. What will be the account value after five years if the payments are made at the end of each year? What will be the account value after five years if the payments are made at the beginning of each year?$7.(5$ points$)$ Tina takes out a loan of $$10,000$ at an annual interest rate of $6\%,$ compounded quarterly. She plans to repay the loan in equal quarterly payments of $$600.$ How long will it take Tina to repay the loan fully?$8.(5$ points$)$ Samantha wants to save $$15,000$ for a new car in $4$ years. If she can earn an interest rate of $5$ percent, compounded annually, how much does Samantha need to deposit today to have the $$15,000$ in $4$ years?